Universal method for precise projectile flight prediction

ABSTRACT

Providing radar or a sensor compatibility with the proposed device, their performance will be enormously improved by enabling them to point at exact search, rescue, or evacuation area ahead of time.

U.S. PATENT DOCUMENTS

3,748,440 July 1973 Alexander 235/61.5 R 6,262,680 B1 June 2001 Muto September 1999 382/103 7,605,747 B1 October 2006 Mookerjee et al. 342/90

OTHER PUBLICATIONS

-   Burnett B. “Trajectory Equations for A Six_Degree_of_Freedom     Missile”, FRL-TM-25, May 1962. FAA, “Coordinate Conversion”, I-295     to I-312, DTFA01-88-c-00042, CDRL-EN25, Change 2, Volume I, 6 Sep.     1991. -   Goloubev M., Shimizu M. “Process of Predicting Geographic     Coordinates of Projectile Impact under Constant Angular Momentum”,     US Pending Patent, EFS ID 20677495, 2014 -   Isaacson J., etc., “Estimation and prediction of ballistic missile     trajectories”, Project AIR FORCE, RAND/MR-737-AF, 1996 -   Kashiwagi Y. “Prediction of ballistic missile trajectories”,     Memorandum37, AD728502, SRI Project 5188-305, June 1968. -   Siouris G. “Missile Guidance and Control Systems”, Springer, p 666,     XIV, 2004 -   “The effects of the Coriolis force on projectile trajectories”,     University of British Columbia,     https://www.phas.ubc.ca/˜berciu/TEACHING/PHYS206/LECTURES/FILES/coriolis.pdf” -   “Theory of projectile motion”,     http://www.marts100.com/projectile.htm

BACKGROUND OF INVENTION

The precise estimate of a projectile trajectory parameters expected in military applications, space travel, rescue and recovery missions, evacuation warnings, games and hobbies, etc. is a challenging task requiring exact analysis of the three-dimensional projectile flight. Currently existing to it solutions are rather complicated and therefore inconvenient. As cited in U.S. Pat. No. 3,748,440 solutions to two dimensional non-linear differential equations are developed in inertial coordinate systems where integrations are performed to obtain X and Y coordinates associated with Latitude and Longitude. Geometric Line of Sight angles is used in U.S. Pat. No. 6,262,680 B1 to track the target in inertial coordinate system. In U.S. Pat. No. 7,605,747 B1 position and velocity vectors are referenced to a non-inertial reference frame such as Earth Centered Earth Fixed (ECEF) when positional registration bias state vector δX represents the sensor position with respect to the ECEF coordinates. Other publications include introduction of either extra degrees of freedom (Burnett, 1962) in order to describe motion in orthogonal two dimensional planes, or consideration of specific initial conditions (Kashiwagi, 1968), or synchronous geo-satellite (Isaacson et al., 1996) utilizing Kalman Filter algorithm, or algorithm (Siouris, 2004) associated with projectile coordinate along its track. Simple solutions from two technical articles found in the websites are just referring to the Coriolis force to proximate the deviation from estimated locations (https://www.phas.ubc.ca/˜berciu/TEACHING/PHYS206/LECTURES/FILES/coriolis.pdf and http://www.marts100.com/projectile.htm).

Lastly, programmed in MATLAB the process of reverse conversion of Sensor Target Measurements into ECEF coordinates allows predicting Sensor Target Measurements at impact location of projectile and enables the radar to direct its beam to the location (I-295 to I-312, DTFA01-88-c-00042, CDRL-EN25, Change 2, Volume I, 6 Sep. 1991).

General and precise solution to the complex problem of a projectile motion would create the grounds for designing a new device highly efficient for any application related to a projectile flight on any planet.

SUMMARY OF INVENTION

Proposed a new device allowing easily, quickly and accurately predict dependable on time location and speed of a capsule or it debris on any planet: Earth, Mars and Moon included in this application. The working principle of this device is based on developed unique and efficient method to evaluate precise geographical coordinates and speed of a projectile at any moment of its flight for a given launching conditions which could be input manually or transferred from a radar.

Manual input provides with exact parameters of impact allowing to evaluate planned launch efficiency and estimate possible impact outcome, while use of initial radar data predicts projectile flight ahead of time significantly improving radar functionality by enabling to rapidly redirect the radar beam to estimated impact location ahead of impact time.

Ability to use initial radar report provides device with the option of accounting on the mechanical distortions on a projectile flight such as projectile rotation and planet atmospheric conditions resulting in drag force.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1. Solution setup: intermediate projectile position and initial conditions relatively to Launching Horizontal Platform. ECEF Coordinate system i-j-k is replaced by spherical coordinates u-w-v. For convenience of this and further illustrations another launching angle

$\vartheta_{o}^{\prime} = {\frac{\pi}{2} - \vartheta_{o}}$ is shown.

FIG. 2. 2-D solution of a projectile motion in rotating ECEF local coordinates I-J (trace 2-3) coupled to changing longitude and latitude of the spherical triangle centered at the origin.

FIG. 3. Flow Chart Diagram for precise projectile flight prediction using either manual (sheet 1) or radar provided (sheet 2) input and common output (sheet 3).

DETAILED DESCRIPTION A: Definitions (FIG. 1)

The following definitions are useful in understanding the process of precise projectile aiming and tracing for rapidly directing the radar sensor beam to geographical impact location of objects.

-   -   Ω is a vector of angular velocity of rotating planet pointing         from South-to-North;     -   γ is universal gravitational constant;     -   M is mass of the planet;     -   r, r_(o), r₁ are respectively intermediate, launching and next         to launching radial distance of a projectile measured from the         planet center considered to be the origin of reference frame;     -   φφ_(o), φ₁ are correspondingly intermediate, initial and next to         launching position projectile altitudinal angle in chosen frame         of reference;     -   θ, θ_(o), θ₁ are respectively running, launching and next to         launching position projectile azimuthal angle in chosen         reference frame;     -   _(o) is projectile launching angle measured from launching         horizontal platform. It is positive when measured in         counterclockwise direction;     -   V_(o) is projectile launching velocity;     -   V_(ro), V_(θo) and V_(φ0) are radial, azimuthal and altitudinal         components of launching velocity correspondingly     -   β is bearing angle (direction of launch). It is positive when         measured counterclockwise from south direction (i.e. from south-         to east- to north);     -   L² is the square of magnitude of constant total projectile         angular momentum

B: Development of Relevant Solutions

When projectile velocity way below speed of light, unique solutions for its motion can be developed through application of classical mechanics:

According to the Second Newton's Law general equation of projectile dynamics in non-inertial frame of reference, which rotates with constant angular velocity Ω, is

$\begin{matrix} {{{{{- \gamma}\frac{M}{r^{3}}r} - {{2 \cdot \Omega} \times V} - {\Omega \times \left( {\Omega \times r} \right)}} = \frac{d\overset{\rightarrow}{V}}{dt}},} & \left( {{Eq}.\mspace{14mu} 1} \right) \end{matrix}$ where

${- \gamma}\frac{mM}{r^{3}}r$ is gravitational force;

-   -   2·δ×V is Coriolis force;     -   Ω×(Ω×r) is centrifugal force.

Introduction of radial unit vector u and its derivatives:

u = sin  φ ⋅ cos  θ ⋅ i + sin  φ ⋅ sin  θ ⋅ j + cos  φ ⋅ k; u² = 1 ${\frac{\partial u}{\partial\varphi} = {w = {{\cos\;{\varphi \cdot \cos}\;{\theta \cdot i}} + {\cos\;{\varphi \cdot \sin}\;{\theta \cdot j}} - {\sin\;{\varphi \cdot k}}}}};{w^{2} = 1}$ $\frac{\partial w}{\partial\varphi} = {{- u} = {{{- \sin}\;{\varphi \cdot \cos}\;{\theta \cdot i}} - {\sin\;{\varphi \cdot \sin}\;{\theta \cdot j}} - {\cos\;{\varphi \cdot k}}}}$ ${\frac{\partial u}{\partial\theta} = {v = {{{- \sin}\;{\varphi \cdot \sin}\;{\theta \cdot i}} + {\sin\;{\varphi \cdot \cos}\;{\theta \cdot j}} + {0 \cdot k}}}};{v^{2} = {{\sin^{2}\varphi\frac{\partial v}{\partial\theta}} = {\sigma = {{{{- \sin}\;{\varphi \cdot \cos}\;{\theta \cdot i}} - {\sin\;{\varphi \cdot \sin}\;{\theta \cdot j}} + {{0 \cdot k}\frac{\partial v}{\partial\varphi}}} = {\frac{\partial w}{\partial\theta} = {\delta = {{{{- \cos}\;{\varphi \cdot \sin}\;{\theta \cdot i}} + {\cos\;{\varphi \cdot \cos}\;{\theta \cdot j}} + {{0 \cdot k}{du}}} = {{{{\frac{\partial u}{\partial\varphi} \cdot d}\;\varphi} + {{\frac{\partial u}{\partial\theta} \cdot d}\;\theta}} = {{{{w \cdot d}\;\varphi} + {{v \cdot d}\;\theta{dw}}} = {{{{\frac{\partial w}{\partial\varphi} \cdot d}\;\varphi} + {{\frac{\partial w}{\partial\theta} \cdot d}\;\theta}} = {{{{{- u} \cdot d}\;\varphi} + {{\delta \cdot d}\;\theta{dv}}} = {{{{\frac{\partial v}{\partial\varphi} \cdot d}\;\varphi} + {{\frac{\partial v}{\partial\theta} \cdot d}\;\theta}} = {{{\delta \cdot d}\;\varphi} + {{\sigma \cdot d}\;\theta}}}}}}}}}}}}}}$ as well as taking into consideration that r=r·u and

${V = {\frac{dr}{dt} = {{\overset{.}{r} \cdot u} + {r \cdot \overset{.}{\varphi} \cdot w} + {r \cdot \overset{.}{\theta} \cdot v}}}},$ where altitudinal angular speed

$\overset{.}{\varphi} = \frac{d\;\varphi}{dt}$ while azimuthal angular speed

${\overset{.}{\theta} = \frac{d\;\theta}{dt}},$ modifies Eq.1 as:

${{\left( {{{- \gamma}\frac{M}{r^{2}}} + {\Omega^{2}r}} \right) \cdot u} - {{2 \cdot \Omega} \times V} - {r\;{\Omega \cdot \left( {\Omega \cdot u} \right)}}} = {{\left( {\overset{¨}{r} - {r\;{\overset{.}{\varphi}}^{2}}} \right)u} + {\left( {{2\overset{.}{r}\overset{.}{\varphi}} + {r\;\overset{¨}{\varphi}}} \right)w} + {\left( {{2\overset{.}{r}\overset{.}{\theta}} + {r\;\overset{¨}{\theta}}} \right)v} + {2r\;\overset{.}{\varphi}\overset{.}{\theta}\delta} + {r\;{\overset{.}{\theta}}^{2}\sigma}}$ r{dot over (θ)}²σ Consequent dot multiplication of this equation first by v, then by w and u correspondingly provides with:

$\begin{matrix} {\frac{dr}{r} = {{- \frac{d\left( {\sin\;\varphi} \right)}{\sin\;\varphi}} - \frac{d\;\overset{.}{\theta}}{2 \cdot \left( {\Omega + \overset{.}{\theta}} \right)}}} & \left( {{Eq}.\mspace{14mu} 2} \right) \\ {{\left\lbrack \frac{\overset{.}{\varphi}}{{\left( {\overset{.}{\theta} + \Omega} \right) \cdot \sin^{2}}\varphi} \right\rbrack \cdot {d\left\lbrack \frac{\overset{.}{\varphi}}{{\left( {\overset{.}{\theta} + \Omega} \right) \cdot \sin^{2}}\varphi} \right\rbrack}} = \frac{d\left( {\sin\mspace{14mu}\varphi} \right)}{\sin^{3}\varphi}} & \left( {{Eq}.\mspace{14mu} 3} \right) \\ {{{\overset{.}{r}d\overset{.}{r}} - {{r\left\lbrack {{\overset{.}{\varphi}}^{2} + {\left( {\overset{.}{\theta} + \Omega} \right)^{2}\sin^{2}\varphi}} \right\rbrack} \cdot {dr}} + {\gamma{\frac{M}{r^{2}} \cdot {dr}}}} = 0} & \left( {{Eq}.\mspace{14mu} 4} \right) \end{matrix}$

Solution of Eqs.2 and 3 leads to the conclusion that r⁴ [{dot over (φ)}²+({dot over (θ)}+Ω)² sin² φ]=const

This constant is actually squared magnitude of projectile angular momentum: L ² =r ⁴[{dot over (φ)}²+({dot over (θ)}+Ω)² sin²φ]=const  (Eq.5) which could be defined as L=r ²[{dot over (φ)}w+({dot over (θ)}+Ω)v]  (Eq.6)

Altitudinal component L_(φ)={dot over (φ)}w of angular momentum in not affected by rotational frame of reference and is the same as it would be in inertial frame, while azimuthal component L_(θ) consists of two coaxial vectors: angular momentum L′_(θ)={dot over (θ)}v projectile would have in inertial frame and additional component L_(Ω)=Ωv due to reference frame rotation. Magnitude of angular momentum remains constant, but its direction is constantly changing so its rotating vector creates conic surface.

Substitution Eq.5 into Eq.4 reveals energy −e conservation in non-inertial frame of reference:

$\begin{matrix} {{\frac{{\overset{.}{r}}^{2}}{2} + \frac{L^{2}}{2 \cdot r^{2}} - {\gamma\frac{M}{r}}} = {{\frac{V^{2}}{2} - {\gamma\frac{M}{r}}} = {- e}}} & \left( {{Eq}.\mspace{14mu} 7} \right) \end{matrix}$

Vector of projectile velocity V contains radial V_(r)={dot over (r)}u, altitudinal V_(φ)=r{dot over (φ)}w, and azimuthal V_(θ)=r({dot over (θ)}+Ω)v components and can be presented as V={dot over (r)}u+rω, where rotational velocity in the projectile plane

$\omega = {{{\overset{.}{\varphi}\omega} + {\left( {\overset{.}{\theta} + \Omega} \right)v}} = {\frac{L}{r^{2}}.}}$ Magnitude of ω is defined by rotational angle measured in this plane (FIG. 2):

$\begin{matrix} {\omega = \frac{d\;\vartheta}{dt}} & \left( {{Eq}.\mspace{14mu} 8} \right) \end{matrix}$

Magnitude of angular momentum L² and projectile total energy −e could be defined from initial condition as

$\begin{matrix} {L^{2} = {r_{o}^{2} \cdot \left( {V_{\varphi_{o}}^{2} + V_{\theta_{o}}^{2} + {2r_{o}^{2}{\overset{.}{\theta}}_{o}{\Omega \cdot \sin^{2}}\varphi_{o}} + {r_{o}^{2}{\Omega^{2} \cdot \sin^{2}}\varphi_{o}}} \right)}} & \left( {{Eq}.\mspace{14mu} 9} \right) \\ {e = {{\gamma\frac{M}{r_{o}}} - \frac{V_{ro}^{2}}{2} - \frac{L^{2}}{2r_{o}^{2}}}} & \left( {{Eq}.\mspace{14mu} 10} \right) \end{matrix}$

Here V_(ro)={dot over (r)}_(o)=V_(o) sin ϑ_(o); V_(φ) _(o) =r_(o){dot over (φ)}_(o)=V_(o) cos ϑ_(o)·cos(±β); and V_(θ) _(o) =r_(o){dot over (θ)}_(o) sin φ_(o)=V_(o) cos ϑ_(o)·sin(±β) are respectively radial, altitudinal and azimuthal components of velocity in inertial (stationary) frame of reference;

r_(o), V_(o), and ϑ_(o) are launching radial coordinate, velocity and angle correspondingly; β is horizontal direction of launch negative for westward and positive for eastward measurement (FIG. 1)

Eq.10 provides with the expression for maximum projectile elevation when its radial velocity is zero:

$\begin{matrix} {r_{\max} = \frac{{\gamma\; M} + \sqrt{K}}{2e}} & \left( {{Eq}.\mspace{14mu} 11} \right) \end{matrix}$ where flight constant K=γ² M²−2eL²

Substitution Eq.10 into Eq.7 reveals

$\begin{matrix} {\frac{dr}{dt} = {\frac{1}{r} \cdot \sqrt{{{- 2}{e \cdot r^{2}}} + {2\gamma\;{M \cdot r}} - L^{2}}}} & \left( {{Eq}.\mspace{14mu} 12} \right) \end{matrix}$ with solution for flight time t:

$\begin{matrix} {t = \left\{ \begin{matrix} {{t_{a} = {\frac{\sqrt{f\left( r_{o} \right)} - \sqrt{f(r)}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}} - {\gamma\; M}}{\sqrt{K}}} - {\arcsin\frac{{2{er}_{o}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},{r \geq r_{o}}} \\ {{t_{\max} = {\frac{\sqrt{f\left( r_{o} \right)} - \sqrt{f\left( r_{\max} \right)}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{\max}} - {\gamma\; M}}{\sqrt{K}}} - {\arcsin\frac{{2{er}_{o}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},{r = r_{\max}}} \\ {{t_{d} = {t_{\max} + \frac{\sqrt{f(r)} - \sqrt{f\left( r_{\max} \right)}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{\max}} - {\gamma\; M}}{\sqrt{K}}} - {\arcsin\frac{{2{er}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},{r < r_{\max}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 13} \right) \end{matrix}$

-   -   where         f(r _(o))=−2er _(o) ²+2γMr _(o) −L ²;         f(r _(max))=−2er _(max) ²+2γMr _(max) −L ²;         f(r)=−2er ²+2γMr−L ²;

Substitution of Eq.2 solution: (Ω+{dot over (θ)})·r ²·sin² φ=L _(θ)=const into Eq.5 leads to

$\begin{matrix} {{\sin\mspace{14mu}\varphi\mspace{14mu} d\;\varphi} = {\frac{\sqrt{{L^{2}\sin^{2}\varphi} - L_{\theta}^{2}}}{r^{2} \cdot \overset{.}{r}}{dr}}} & \left( {{Eq}.\mspace{14mu} 14} \right) \end{matrix}$ providing with altitudinal coordinate φ:

$\begin{matrix} {\varphi = \left\{ \begin{matrix} {{\varphi_{a} = {\varphi_{o} + {\frac{\cos\mspace{14mu}\beta}{\left| {\cos\mspace{14mu}\beta} \right|}{\arcsin\left\lbrack {C \cdot R_{a}} \right\rbrack}}}},{r \geq r_{o}}} \\ {{\varphi_{\max} = {\varphi_{o} + {\frac{\cos\mspace{14mu}\beta}{\left| {\cos\mspace{14mu}\beta} \right|}{\arcsin\left\lbrack {C \cdot R_{\max}} \right\rbrack}}}},{r = r_{\max}}} \\ {{\varphi_{d} = {\varphi_{\max} + {\frac{\cos\mspace{14mu}\beta}{\left| {\cos\mspace{14mu}\beta} \right|}{\arcsin\left\lbrack {C \cdot R_{d}} \right\rbrack}}}},{r < r_{\max}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 15} \right) \end{matrix}$ where

${C = {\sqrt{\frac{{\overset{.}{\varphi}}_{o}^{2} + {\left( {{\overset{.}{\theta}}_{o} + \Omega} \right)^{2}\sin^{2}\varphi_{o}\cos^{2}\varphi_{o}}}{{\overset{.}{\varphi}}_{o}^{2} + {\left( {{\overset{.}{\theta}}_{o} + \Omega} \right)^{2}\sin^{2}\varphi_{o}}}} = \sqrt{\frac{{{r_{o}^{2} \cdot V_{\varphi\; o}^{2} \cdot \sin^{2}}\varphi_{o}} + {{L^{2} \cdot \cos^{2}}\varphi_{o}}}{L^{2}}}}},$ function

$\begin{matrix} {R = \left\{ \begin{matrix} {{R_{a} = {L \cdot \frac{{\left( {{\gamma\;{Mr}} - L^{2}} \right)\sqrt{f\left( r_{o} \right)}} - {\left( {{\gamma\;{Mr}_{o}} - L^{2}} \right)f\sqrt{(r)}}}{K \cdot r \cdot r_{o}}}},{r \geq r_{o}}} \\ {{R_{\max} = {L \cdot \frac{{\left( {{\gamma\;{Mr}_{\max}} - L^{2}} \right)\sqrt{f\left( r_{o} \right)}} - {\left( {{\gamma\;{Mr}_{o}} - L_{2}} \right)\sqrt{f\left( r_{\max} \right)}}}{K \cdot r_{o} \cdot r_{\max}}}},{r = r_{\max}}} \\ {{R_{d} = {L \cdot \frac{{\left( {{\gamma\;{Mr}_{\max}} - L^{2}} \right)\sqrt{f(r)}} - {\left( {{\gamma\;{Mr}} - L^{2}} \right)\sqrt{f\left( r_{\max} \right)}}}{K \cdot r \cdot r_{o}}}},{r < r_{\max}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 16} \right) \end{matrix}$ and coefficient

$\frac{\cos\;\beta}{\left| {\cos\;\beta} \right|}$ indicates altitudinal coordinate increase or decrease depending on the launching direction β.

Substituting expression for time increment dt expressed in terms of Eq.12 into Eq.8 obtain

$\begin{matrix} {{d\;\vartheta} = {L \cdot \frac{dr}{r \cdot \sqrt{{{- 2}{e \cdot r^{2}}} + {2\gamma\;{M \cdot r}} - L^{2}}}}} & \left( {{Eq}.\mspace{14mu} 17} \right) \end{matrix}$ with solution for rotational angle

:

$\begin{matrix} {\vartheta = {{\arcsin\lbrack R\rbrack} = \left\{ \begin{matrix} {{\vartheta_{a} = {\arcsin\left\lbrack R_{a} \right\rbrack}},{r \geq r_{o}}} \\ {{\vartheta_{\max} = {\arcsin\left\lbrack R_{\max} \right\rbrack}},{r = r_{\max}}} \\ {{\vartheta_{d} = {\vartheta_{\max} + {\arcsin\left\lbrack R_{d} \right\rbrack}}},{r < r_{\max}}} \end{matrix} \right.}} & \left( {{Eq}.\mspace{14mu} 18} \right) \end{matrix}$

Employment of the latest equation (Eq.8) together with Eqs.6 and 15 reveals:

$\begin{matrix} {\frac{d\;\varphi}{\left( {{d\;\theta} + {\Omega\;{dt}}} \right)\mspace{14mu}\sin\mspace{14mu}\varphi} = {\frac{C}{\sqrt{1 - C^{2}}} \cdot \sqrt{1 - R^{2}}}} & \left( {{Eq}.\mspace{14mu} 19} \right) \end{matrix}$ with the solution for azimuthal coordinate θ:

$\begin{matrix} {\theta = \left\{ \begin{matrix} {{\theta_{a} = \left. {\theta_{o} + {{\frac{\sin\mspace{14mu} 2\beta}{\left| {\sin\mspace{14mu} 2\beta} \right|} \cdot \frac{\sqrt{1 - C^{2}}}{C\sqrt{1 - R_{a}^{2}}}}\ln}} \middle| \frac{\tan\left( {\varphi_{a}\text{/}2} \right)}{\tan\left( {\varphi_{o}\text{/}2} \right)} \middle| {{- \Omega} \cdot t_{a}} \right.},{r \geq r_{o}}} \\ {{\theta_{\max} = {\theta_{o} + {{\frac{\sin\mspace{14mu} 2\beta}{\left| {\sin\mspace{14mu} 2\beta} \right|} \cdot \frac{\sqrt{1 - C^{2}}}{C\sqrt{1 - R_{\max}^{2}}}}\ln\frac{\left| {\tan\left( {\varphi_{\max}\text{/}2} \right)} \right|}{\left| {\tan\left( {\varphi_{o}\text{/}2} \right)} \right|}} - {\Omega \cdot t_{\max}}}},{r = r_{\max}}} \\ {{\theta_{d} = {\theta_{\max} + {{\frac{\sin\mspace{14mu} 2\beta}{\left| {\sin\mspace{14mu} 2\beta} \right|} \cdot \frac{\sqrt{1 - C^{2}}}{C\sqrt{1 - R_{d}^{2}}}}\ln\frac{\left| {\tan\left( {\varphi_{d}\text{/}2} \right)} \right|}{\left. {\tan\left( {\varphi_{\max}\text{/}2} \right)} \right|}} - {\Omega \cdot \left( {t_{d} - t_{\max}} \right)}}},{r < r_{\max}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 20} \right) \end{matrix}$ where coefficient

$\frac{\sin\mspace{14mu} 2\beta}{\left| {\sin\mspace{14mu} 2\beta} \right|}$ takes into account azimuthal coordinate increase or decrease depending on the launching direction β.

Eqs.13, 15, 16, and 20 is actually the parametric system of equations defining a projectile trajectory in non-inertial frame of reference and require for their completion equation of hodograph, i.e. velocity vs. time dependence.

Due to energy conservation (Eq.8) projectile speed at any moment of time can be defined as

$\begin{matrix} {V = \sqrt{{{V_{o}^{2} \cdot \sin^{2}}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2\gamma\; M}{r_{o}r}\left( {r - r_{o}} \right)}}} & \left( {{Eq}.\mspace{14mu} 21} \right) \end{matrix}$

Introduction of projectile ascending speed V_(a), speed at maximum elevation V_(min), and descending speed V_(d) breaks Eq.21 into three parts logically fitting trajectory analysis:

$\begin{matrix} {V = \left\{ \begin{matrix} {{V_{a} = \sqrt{{V_{o}^{2}\sin^{2}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2\gamma\; M}{r_{o}r}\left( {r - r_{o}} \right)}}},{r \geq r_{o}}} \\ {{V_{\min} = \sqrt{{V_{o}^{2}\sin^{2}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2\gamma\; M}{r_{o}r_{\max}}\left( {r_{\max} - r_{o}} \right)}}},{r = r_{\max}}} \\ {{V_{d} = \sqrt{V_{\min}^{2} + {\frac{2\gamma\; M}{r_{\max}r}\left( {r_{\max} - r} \right)}}},{r < r_{\max}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 22} \right) \end{matrix}$

Derived above equations of projectile motion in three-dimensional non-inertial frame of reference indicate that, due to the presence of additional component of angular momentum, projectile coordinates are going to be shifted in westward direction “twisting” the trajectory plane.

The other obvious reasons for coordinates shift is possible projectile rotation, atmospheric conditions, drag force etc. which are actually taken into account by a radar.

Actually, any point of projectile trajectory can be treated as the “launching” (with script 0) one. Next to it position (with script 1) traced by a radar is so close to the initial that both projectile positions can be assumed to be in flat two dimensional plane. Thus, these two radar records can be treated as a projectile moving in inertial frame of reference (Ω=0) under existing deflecting its flight conditions.

Equation of projectile dynamics according to Eq.1 in this case becomes

$\begin{matrix} {{{- \gamma}\frac{M}{r^{3}}r} = \frac{d\overset{\rightarrow}{V}}{dt}} & \left( {{Eq}.\mspace{14mu} 23} \right) \end{matrix}$

Introduction (FIG. 2) of radial unit vector U and its derivatives:

U = cos   ϑ ⋅ I + sin   ϑ ⋅ J $W = {\frac{dU}{d\;\vartheta} = {{{- \sin}\mspace{14mu}{\vartheta \cdot I}} + {\cos\mspace{14mu}{\vartheta \cdot J}}}}$ $\frac{dW}{d\;\vartheta} = {{{{- \cos}\mspace{14mu}{\vartheta \cdot I}} - {\sin\mspace{14mu}{\vartheta \cdot J}}} = {- U}}$ and taking into consideration that r=r·U;

${V = {\frac{dr}{dt} = {{\overset{.}{r} \cdot U} + {r \cdot \overset{.}{\vartheta} \cdot W}}}};$ and L=r²

=r_(o)·V_(o)·cos

_(o) provides with the solution:

$\begin{matrix} {r = \frac{L^{2}}{\gamma\;{M\left\lbrack {1 - {\cos\mspace{14mu}\vartheta} + {\frac{L}{\gamma\; M}V_{o}\mspace{14mu}{\cos\left( {\vartheta_{o} + \vartheta} \right)}}} \right\rbrack}}} & \left( {{Eq}.\mspace{14mu} 24} \right) \end{matrix}$ which, in turn, reveals the following expressions for two consequent projectile positions:

$r_{o} = {\frac{L}{{V_{o} \cdot \cos}\mspace{14mu}\vartheta_{o}}\mspace{14mu}{and}}$ ${r_{1} = \frac{L^{2}}{\gamma\;{M\left\lbrack {1 - {\cos\mspace{14mu}\vartheta_{1}} + {\frac{L}{\gamma\; M}V_{o}\mspace{14mu}{\cos\left( {\vartheta_{o} + \vartheta_{1}} \right)}}} \right\rbrack}}},$ where, cos

₁=cos(φ₁−φ_(o))·cos [(θ₁−θ_(o))·sin φ₁] what is the cosine rule of spherical right triangle centered at the origin of spherical coordinates (FIG. 2).

If the set of longitude/latitude radar coordinates is Lo0/La0 and Lo1/La1 then altitudinal and azimuthal angles are defined as follows: θ₀=Lo0; θ₁=Lo1; φ₀=90°−La0; φ₁=90°−La1 Application of energy conservation to radar provided positions:

${\frac{{V_{o}^{2} \cdot \sin^{2}}\vartheta_{o}}{2} + \frac{L^{2}}{2r_{o}^{2}} - \frac{\gamma\; M}{r_{o}}} = {\frac{V_{1}^{2} \cdot {\sin^{2}\left( {\vartheta_{o} + \vartheta_{1}} \right)}}{2} + \frac{L^{2}}{2r_{1}^{2}} - \frac{\gamma\; M}{r_{1}}}$ reveals the solution for initial (launching) angle

$\begin{matrix} {\vartheta_{o} = {\arctan\left\lbrack \frac{{r_{1}V_{1}\cos\;\vartheta_{1}} - r_{{oV}_{o}}}{r_{1}V_{1}\sqrt{1 - {\cos^{2}\vartheta_{1}}}} \right\rbrack}} & \left( {{Eq}.\mspace{14mu} 25} \right) \end{matrix}$ positive for ascending projectile and negative for descending.

Direction of “launch” can be defined from the sine rule of spherical right triangle (FIG. 2) as

$\begin{matrix} {\beta = {\arcsin\left\{ \frac{\sin\left\lbrack {{\left( {\theta_{1} - \theta_{o}} \right) \cdot \sin}\mspace{14mu}\varphi_{1}} \right\rbrack}{\sqrt{1 - {\cos^{2}\vartheta_{1}}}} \right\}}} & \left( {{Eq}.\mspace{14mu} 26} \right) \end{matrix}$

Establishing “launching” parameters of a projectile flight from two consequent radar measurements allows to further predict projectile real behavior ahead of time.

Developed technique creates the basis of device operation (FIG. 3) in two modes using either manual or provided by a radar input, and both allowing predict exact trajectory of a projectile and its speed at impact.

TABLE Flow Chart Diagram for Precise Projectile Flight Prediction: Predefined Data for: Planet Earth Gravitational constant γ in preferred system of units $\gamma = {6.674 \times 10^{- 11}\frac{m^{3}}{{kg}*s^{2}}}$ Planet radius r_(P) measured in the same system of units r_(P) = 6.37 × 10⁶ m Planet mass M measured in the same system of units M =5.972 × 10²⁴ kg Planet rotational speed Ω measured in radians per unit time Ω = 7.29 × 10⁻⁵ s⁻¹ Planet Moon Gravitational constant γ in preferred system of units $\gamma = {6.674 \times 10^{- 11}\frac{m^{3}}{{kg}*s^{2}}}$ Planet radius r_(P) measured in the same system of units r_(P) = 1.74 × 10⁶ m Planet mass M measured in the same system of units M = 7.35 × 10²² kg Planet rotational speed Ω measured in radians per unit time Ω = 2.67 × 10⁻⁶ s⁻¹ Planet Mars Gravitational constant γ in preferred system of units $\gamma = {6.674 \times 10^{- 11}\frac{m^{3}}{{kg}*s^{2}}}$ Planet radius r_(P) measured in the same system of units r_(P) = 3.39 × 10⁶ m Planet mass M measured in the same system of units M = 6.39 × 10²³ kg Planet rotational speed Ω measured in radians per unit time Ω = 7.09 × 10⁻⁵ s⁻¹ Planet choice provides with:  2: Value γ · M  3: Value r_(p)  4: Value Ω; this vector is always pointing from South Pole - to - North Pole Manual Input:  5: Enter launching direction angle β (in °) positive when measured on horizontal launching platform counterclockwise from direction of south-to-east-to-north (SEN) and negative for south-to-west-to-north (SWN) direction thus, defined as β(1 − 2 · ba), where ${ba} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{SEN}} \\ {1\mspace{14mu}{for}\mspace{14mu}{SWN}} \end{matrix} \right.$  6: Enter launching angle ϑ_(o) (in °) positive when measured in upward (counterclockwise) direction (UpD) from launching horizontal platform and negative for downward direction (DwD) hence, defined as ϑ_(o) (1 − 2 · ud), where ${ud} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{UpD}} \\ {1\mspace{14mu}{for}\mspace{14mu}{DwD}} \end{matrix} \right.$  7: Enter launching speed V_(o) (in m/sec)  8: Enter Launching Altitude above planet Surface Level LASL (in meters) positive when above and negative when under surface level thus, radial launching coordinate r_(o) is computed as r_(o) = r_(p) + LASL  9: Enter Launching Latitude LLa (in °) positive for northern hemisphere (NHS) and negative for southern (SHS) thus, defined as LLa(1 − 2 · ns), where ${ns} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{NHS}} \\ {1\mspace{14mu}{for}\mspace{14mu}{SHS}} \end{matrix} \right.$ 10: Enter Launching Longitude LLo (in °) positive in eastern hemisphere (EHS) and negative in western (WHS) hence, defined as LLo(1 − 2 · ew), where ${ew} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{EHS}} \\ {1\mspace{14mu}{for}\mspace{14mu}{WHS}} \end{matrix} \right.$ 11: Enter Impact Altitude above planet Surface Level IASL (in meters) positive when above and negative when below surface level hence, radial impact coordinate r_(i) is computed as r_(i) = r_(p) + IASL Flight Constants Computation: 12: Initial Radial Speed V_(ro) = V_(o) sin(ϑ_(o)) 13: Initial Altitudinal Speed V_(φ) _(o) = V_(o) cos(ϑ_(o)) · cos(β) 14: Initial Azimuthal Speed V_(θo) = V_(o)cos(ϑ_(o)) · sin(β) + r_(o) · Ω · cos (LLa) 15: Angular momentum L = r_(o) · {square root over (V_(φo) ² + V_(θo) ²)} 16: Altitudinal coefficient $C = \sqrt{\frac{V_{\varphi\; o}^{2} + {\cdot V_{\theta\; o}^{2} \cdot {\sin^{2}({LLa})}}}{V_{\varphi\; o}^{2} + {\cdot V_{\theta\; o}^{2}}}}$ 17: Total Energy Magnitude $e = {{\gamma\frac{M}{r_{o}}} - \frac{V_{ro}^{2}}{2} - \frac{L^{2}}{2r_{o}^{2}}}$ 18: Flight Constant K = γ²M² − 2eL² 19: Maximum Elevation $r_{{ma}\; x} = \left\{ \begin{matrix} \frac{{\gamma\; M} + \sqrt{K}}{2e} & {{{if}\mspace{14mu}\vartheta_{o}} > 0} \\ r_{o} & {{{if}\mspace{14mu}\vartheta_{o}} \leq 0} \end{matrix} \right.$ Trajectory Parameters for Upward Launch ϑ_(o) > 0: Ascending leg 20: Radial position r_(a) is defined by running numbers from r_(o) to r_(i) through default increment +1m r_(a) = r_(o): r_(i): (+1m) computing: 23: Ascending R-function $R_{a} = {L \cdot \frac{{\left( {{\gamma\;{Mr}_{a}} - L^{2}} \right)\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}} - {\left( {{\gamma Mr}_{o} - L^{2}} \right)\sqrt{{{- 2}{er}_{a}^{2}} + {2\gamma\;{Mr}_{a}} - L^{2}}}}{r_{a} \cdot r_{o} \cdot K}}$ 28: Ascending time $t_{a} = {\frac{\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}} - \sqrt{{{- 2}{er}_{a}^{2}} + {2\gamma\;{Mr}_{a}} - L^{2}}}{2e} + {\frac{\gamma M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{a}} - {\gamma M}}{\sqrt{K}}} - {\arcsin\frac{{2{er}_{o}} - {\gamma M}}{\sqrt{K}}}} \right)}}$ 33: Ascending Latitude ${ALa} = {{LLa} - {\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {\arcsin\left\lbrack {C \cdot R_{a}} \right\rbrack}}}$ 34: Ascending Longitude ${ALo} = {{LLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{a}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{ALa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{LLa}{2}} \right)}}} - {\Omega \cdot t_{a}}}$ 37: Ascending Speed $V_{a} = \sqrt{{{V_{o}^{2} \cdot \sin^{2}}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2{\gamma M}}{r_{a} \cdot r_{o}} \cdot \left( {r_{a} - r_{o}} \right)}}$ Maximum Elevation 24: R-Function $R_{{ma}\; x} = {L \cdot \frac{\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}{r_{o} \cdot \left( {{2{er}_{{ma}\; x}} - {\gamma M}} \right)}}$ 29: Time to maximum elevation $t_{{ma}\; x} = {\frac{\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}{2e} + {\frac{\gamma M}{2e\sqrt{2e}}\left( {\frac{\pi}{2} - {\arcsin\frac{{2{er}_{o}} - {\gamma M}}{\sqrt{K}}}} \right)}}$ 31: Maximum elevation Latitude ${MLa} = {{LLa} - {\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {\arcsin\left\lbrack {C \cdot R_{{ma}\; x}} \right\rbrack}}}$ 32: Maximum elevation Longitude ${MLo} = {{LLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{{ma}\; x}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{MLa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{LLa}{2}} \right)}}} - {\Omega \cdot t_{{ma}\; x}}}$ 38: Speed at maximum elevation $V_{\min} = \sqrt{{{V_{o}^{2} \cdot \sin^{2}}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2{\gamma M}}{r_{{ma}\; x} \cdot r_{o}} \cdot \left( {r_{{ma}\; x} - r_{o}} \right)}}$ Descending Leg 21: Radial position r_(d) is defined by running numbers from r_(max) to r_(i) through default increment −1m r_(d) = r_(max): r_(i): (−1m) computing: 25: R-Function $R_{d} = {L \cdot \frac{{\left( {{\gamma\;{Mr}_{{ma}\; x}} - L^{2}} \right)\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}}} - {\left( {{\gamma Mr}_{d} - L^{2}} \right)\sqrt{{{- 2}{er}_{{ma}\; x}^{2}} + {2\gamma\;{Mr}_{{ma}\; x}} - L^{2}}}}{r_{{ma}\; x} \cdot r_{d} \cdot K}}$ 30: Descending time $t_{d} = {\frac{\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}} - \sqrt{{{- 2}{er}_{{ma}\; x}^{2}} + {2\gamma\;{Mr}_{{ma}\; x}} - L^{2}}}{2e} + {\frac{\gamma M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{{ma}\; x}} - {\gamma M}}{\sqrt{K}}} - {\arcsin\frac{{2{er}_{d}} - {\gamma M}}{\sqrt{K}}}} \right)}}$ 35: Descending Latitude ${DLa} = {{MLa} - {\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {\arcsin\left\lbrack {C \cdot R_{d}} \right\rbrack}}}$ 36: Descending Longitude ${DLo} = {{MLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{d}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{DLa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{MLa}{2}} \right)}}} - {\Omega \cdot t_{d}}}$ 39: Descending Speed $V_{d} = \sqrt{V_{\min}^{2} + {\frac{2{\gamma M}}{r_{d} \cdot r_{{ma}\; x}} \cdot \left( {r_{{ma}\; x} - r_{d}} \right)}}$ Trajectory Parameters for Downward Launch ϑ_(o) < 0: 22: Radial position r_(d) is defined by running numbers from r_(o) to r_(i) through default increment −1m r_(d) = r_(o): r_(i): (−1m) computing: 25: R-Function $R_{d} = {L \cdot \frac{{\left( {{\gamma\;{Mr}_{o}} - L^{2}} \right)\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}}} - {\left( {{\gamma Mr}_{d} - L^{2}} \right)\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}}{r_{o} \cdot r_{d} \cdot K}}$ 30: Descending time $t_{d} = {\frac{\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}} - \sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}{2e} + {\frac{\gamma M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{o}} - {\gamma M}}{\sqrt{K}}} - {\arcsin\frac{{2{er}_{d}} - {\gamma M}}{\sqrt{K}}}} \right)}}$ 35: Descending Latitude ${DLa} = {{MLa} - {\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {\arcsin\left\lbrack {C \cdot R_{d}} \right\rbrack}}}$ 36: Descending Longitude ${DLo} = {{MLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{d}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{DLa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{MLa}{2}} \right)}}} - {\Omega \cdot t_{d}}}$ 40: Descending Speed $V_{d} = \sqrt{V_{o}^{2} + {\frac{2{\gamma M}}{r_{d} \cdot r_{o}} \cdot \left( {r_{o} - r_{d}} \right)}}$ Print trajectory radial, longitudinal and latitudinal coordinates together with projectile velocities at flight times Plot 3-D trajectory marking time and speed at desired locations Radar Input: Radar Input Module (see reference for ECEF coordinate conversion: I-295 to I-312, DTFA01-88-c-00042, CDRL-EN25, Change 2, Volume I, Sep. 06, 1991):  7: Enter “launching” speed V_(o) (in m/sec) Enter next to “launching” point speed V₁ (in m/sec)  8: Enter “Launching” Altitude above planet surface LAo (in meters) positive when above and negative when under surface level thus, radial launching coordinate r_(o) is computed as r_(o) = r_(p) + LAo Enter next to “Launching” point Altitude above planet surface LA₁ (in meters) positive when above and negative when under surface level thus, radial launching coordinate r₁ is computed as r₁ = r_(p) + LA₁  9: Enter “launching” Latitude La0 (in °) positive for northern hemisphere (NETS) and negative for southern (SHS) thus, defined as LLa = La0(1 − 2 · ns), where ${ns} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{NHS}} \\ {1\mspace{14mu}{for}\mspace{14mu}{SHS}} \end{matrix} \right.$ 10: Enter “launching” Longitude Lo0 (in °) positive in eastern hemisphere (EHS) and negative in western (WHS) hence, defined as LLo = Lo0(1 − 2 · ew), where ${ew} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{EHS}} \\ {1\mspace{14mu}{for}\mspace{14mu}{WHS}} \end{matrix} \right.$ 26: Enter next to “launching” point Latitude La1 (in °) positive for northern hemisphere (NHS) and negative for southern (SHS) thus, defined as La1(1 − 2 · ns), where ${ns} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{NHS}} \\ {1\mspace{14mu}{for}\mspace{14mu}{SHS}} \end{matrix} \right.$ 27: Enter next to “launching” point Longitude Lo1 (in °) positive in eastern hemisphere (EHS) and negative in western (WHS) hence, defined as Lo1(1 − 2 · ew), where ${ew} = \left\{ \begin{matrix} {0\mspace{14mu}{for}\mspace{14mu}{EHS}} \\ {1\mspace{14mu}{for}\mspace{14mu}{WHS}} \end{matrix} \right.$ 11: Enter Impact Altitude above planet surface IA (in meters) positive when above and negative when below surface level hence, radial impact coordinate r_(i) is computed as r_(i) = r_(p) + IA Flight Initials and Constants Computation:  5: “Launching” direction angle $\beta = {\frac{180{^\circ}}{\pi}{\arcsin\left\lbrack \frac{\sin\left\lbrack {\left( {{{Lo}\; 1} - {LLo}} \right) \cdot {\cos\left( {{La}\; 1} \right)}} \right\rbrack}{\sqrt{1 - {{\cos^{2}\left( {{LLa} - {{La}\; 1}} \right)} \cdot {\cos^{2}\left\lbrack {\left( {{{Lo}\; 1} - {LLo}} \right) \cdot {\cos\left( {{La}\; 1} \right)}} \right\rbrack}}}} \right\rbrack}}$  6: “Launching” angle $\vartheta_{o} = {\frac{180{^\circ}}{\pi}{\arcsin\left\lbrack \frac{{r_{1}V_{1}{{\cos\left( {{LLa} - {{La}\; 1}} \right)} \cdot {\cos\left\lbrack {\left( {{{Lo}\; 1} - {LLo}} \right) \cdot {\cos\left( {{La}\; 1} \right)}} \right\rbrack}}} - r_{{oV}_{o}}}{r_{1}V_{1}\sqrt{1 - {{\cos^{2}\left( {{LLa} - {{La}\; 1}} \right)} \cdot {\cos^{2}\left\lbrack {\left( {{{Lo}\; 1} - {LLo}} \right) \cdot {\cos\left( {{La}\; 1} \right)}} \right\rbrack}}}} \right\rbrack}}$ 12: Initial Radial Speed V_(ro) = V_(o) sin(ϑ_(o)) 13: Initial Altitudinal Speed V_(φ) _(o) = V_(o) cos(ϑ_(o)) · cos(β) 14: Initial Azimuthal Speed V_(θo) = V_(o)cos(ϑ_(o)) · sin(β) + r_(o) · Ω · cos (La0) 15: Angular momentum L = ro · {square root over (V_(φo) ² + V_(θo) ²)} 16: Altitudinal coefficient $C = \sqrt{\frac{V_{\varphi\; o}^{2} + {\cdot V_{\theta\; o}^{2} \cdot {\sin^{2}({La0})}}}{V_{\varphi\; o}^{2} + {\cdot V_{\theta\; o}^{2}}}}$ 17: Total Energy Magnitude $e = {{\gamma\frac{M}{r_{o}}} - \frac{V_{ro}^{2}}{2} - \frac{L^{2}}{2r_{o}^{2}}}$ 18: Flight Constant K = γ²M² − 2eL² 19: Maximum Elevation $r_{{ma}\; x} = \left\{ \begin{matrix} \frac{{\gamma\; M} + \sqrt{K}}{2e} & {{{if}\mspace{14mu}\vartheta_{o}} > 0} \\ r_{o} & {{{if}\mspace{14mu}\vartheta_{o}} \leq 0} \end{matrix} \right.$ Then continue with switch 20

DISCUSSION

Features of proposed device include:

-   -   Precise estimate of impact coordinates and speed from the input         of launching position, velocity and direction. The chart below         shows that, while flight times are very close for inertial and         non-inertial frames of reference, mismatch distance between         computed landing points in these frames for various identical         initial conditions is quite substantial even for a ground launch         with pretty moderate velocity.

Flight Time Flight Time Launching Launching Launching Launching in sec. for Ω = in sec. for Ω = Mismatch Altitude, m Speed, m/s Angle ϑ_(o)° Direction β° 0 sec⁻¹ 7.292 * 10⁻⁵ sec⁻¹ Distance, m 30,000 1,000 45 −30 181.65 181.38 436.6 30,000 1,000 45 0 181.65 181.79 1,136.4 30,000 100 45 −30 86.07 86.10 502.4 10,000 1,000 45 −30 73.19 73.06 381.7 10,000 100 45 −30 52.98 53.00 308.6 10,000 10 45 −30 45.91 45.93 316.9 1,000 10 45 −30 15.01 15.02 103.4 0 100 45 −30 14.40 14.41 83.7 0 10 45 −30 1.44 1.44 9.99

-   -   Precise estimate of projectile trajectory and its hodograph.         Conversion of spherical coordinates into geographical ones         allows establishing presented in the Table computational method         for direct prediction of exact projectile path and real speed in         rotating reference frame. Having been based on derived equations         proposed device takes into consideration not only Coriolis and         centrifugal forces but also the change of gravity due to         different projectile altitudes. All requirements for input         angles are listed in the Table too.     -   Radar data input by processing two initial consequent radar         readings leading to predicting the whole true trajectory and         projectile speed under actual conditions of flight including         drag force. Actually, this provides radar with a new, currently         not available function significantly improving radar         performance.     -   Efficient computation. Proposed device is based on method which         does not require any iterations. It is fast and exact as         considers all possible factors resulting in projectile flight.     -   Effectiveness on any planet with or without radar support.

While illustrative embodiment of the invention has been shown and described, numerous variations and alternate embodiments, including eliminating one or more of the steps or elements presented herein, will occur to those skilled in the art. Such variations and alternate embodiments are contemplated and can be made without departing from the spirit and scope of the invention as mentioned in the appended claims. 

The invention claimed is:
 1. A universal method of exact analytical solution of an unguided and unpowered projectile flight on a chosen rotating planet comprising the following steps: inputting of two consequently detected projectile initial (launching L) and next to it radial positions r_(o) and r₁, latitudes LLa and La₁, longitudes LLo and Lo₁, velocities V_(o) and V₁ for determining launching angle $\vartheta_{o} = {{arc}\;{\tan\left\lbrack \frac{{r_{1}{V_{1} \cdot {\cos\left( {{La}_{1} - {LLa}} \right)} \cdot {\cos\left( {{Lo}_{1} - {LLo}} \right)} \cdot {\sin\left( {La}_{1} \right)}}} - {r_{o}V_{o}}}{r_{1}V_{1}\sqrt{1 - {{\cos^{2}\left( {{La}_{1} - {LLa}} \right)} \cdot {\cos^{2}\left( {{Lo}_{1} - {LLo}} \right)} \cdot {\sin^{2}\left( {La}_{1} \right)}}}} \right\rbrack}}$ and bearing angle ${\beta = {{arc}\;\sin\left\{ \frac{\sin\left\lbrack {\left( {{Lo}_{1} - {LLo}} \right) \cdot {\sin\left( {La}_{1} \right)}} \right\rbrack}{\sqrt{1 - {{\cos^{2}\left( {{La}_{1} - {LLa}} \right)} \cdot {\cos^{2}\left( {{Lo}_{1} - {LLo}} \right)} \cdot {\sin^{2}\left( {La}_{1} \right)}}}} \right\}}};$ processing input by defining a projectile flight (in gravitational field of a chosen planet of mass M rotating with angular velocity Ω) constants initial radial speed V_(ro)=V_(o) sin(ϑ_(o)), initial altitudinal speed V_(φo)=V_(o) cos(ϑ_(o))·cos(β), initial azimuthal speed V_(θo)=V_(o) cos(ϑ_(o))·sin(β)+r_(o)·Ω·cos(LLa), angular momentum ${L = {r_{o} \cdot \sqrt{V_{\varphi\; o}^{2} + V_{\theta\; o}^{2}}}},$ altitudinal coefficient ${C = \sqrt{\frac{V_{\varphi\; o}^{2} + {V_{\theta\; o}^{2} \cdot {\sin^{2}({LLa})}}}{V_{\varphi\; o}^{2} + V_{\theta\; o}^{2}}}},$ magnitude of total energy $e = {{\gamma\frac{M}{r_{o}}} - \frac{V_{ro}^{2}}{2} - \frac{L^{2}}{2r_{o}^{2}}}$ (where γ is gravitational constant), flight constant K=γ²M²−2eL², maximum elevation $r_{\max} = \left\{ {\begin{matrix} \frac{{\gamma\; M} + \sqrt{K}}{2e} & {{{if}\mspace{14mu}\vartheta_{o}} > 0} \\ r_{o} & {{{if}\mspace{14mu}\vartheta_{o}} \leq 0} \end{matrix};} \right.$ estimating exact projectile flight parameters for launching angle ϑ_(o)>0: ascending radial coordinate r_(a)=r_(o):r_(i):(+1m), ascending R-function ${R_{a} = {L \cdot \frac{\begin{matrix} {{\left( {{\gamma\;{Mr}_{a}} - L^{2}} \right)\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}} -} \\ {\left( {{\gamma\;{Mr}_{o}} - L^{2}} \right)\sqrt{{{- 2}{er}_{a}^{2}} + {2\;\gamma\;{Mr}_{a}} - L^{2}}} \end{matrix}}{r_{a} \cdot r_{o} \cdot K}}},$ ascending time ${t_{a} = {\frac{\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}} - \sqrt{{{- 2}{er}_{a}^{2}} + {2\;\gamma\;{Mr}_{a}} - L^{2}}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {{{arc}\;\sin\frac{{2{er}_{a}} - {\gamma\; M}}{\sqrt{K}}} - {{arc}\;\sin\frac{{2{er}_{o}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},$ ascending latitude ${{ALa} = {{LLa} - {{\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {arc}}\;{\sin\left\lbrack {C \cdot R_{a}} \right\rbrack}}}},$ ascending longitude ${{ALo} = {{LLo} + {{\frac{\sin\; 2\beta}{{\sin\; 2\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{a}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{ALa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{LLa}{2}} \right)}}} - {\Omega \cdot t_{a}}}},$ ascending speed ${V_{a} = \sqrt{{{V_{o}^{2} \cdot \sin^{2}}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2\gamma\; M}{r_{a} \cdot r_{o}} \cdot \left( {r_{a} - r_{o}} \right)}}},$ R-function at maximum elevation ${R_{\max} = {L \cdot \frac{\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}{r_{o} \cdot \left( {{2{er}_{\max}} - {\gamma\; M}} \right)}}},$ time to maximum elevation ${t_{\max} = {\frac{\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {\frac{\pi}{2} - {{arc}\;\sin\frac{{2{er}_{o}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},$ maximum elevation latitude ${{MLa} = {{LLa} - {{\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {arc}}\;{\sin\left\lbrack {C \cdot R_{\max}} \right\rbrack}}}},$ longitude ${{MLo} = {{LLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{\max}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{MLa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{LLa}{2}} \right)}}} - {\Omega \cdot t_{\max}}}},$ speed at maximum elevation ${V_{\min} = \sqrt{{{V_{o}^{2} \cdot \sin^{2}}\vartheta_{o}} + \frac{L^{2}}{r_{o}^{2}} - {\frac{2\gamma\; M}{r_{\max} \cdot r_{o}} \cdot \left( {r_{\max} - r_{o}} \right)}}},$ descending radial coordinate r_(d)=r_(max):r_(i):(−1m), descending R-function ${R_{d} = {L \cdot \frac{\begin{matrix} {{\left( {{\gamma\;{Mr}_{\max}} - L^{2}} \right)\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}}} -} \\ {\left( {{\gamma\;{Mr}_{d}} - L^{2}} \right)\sqrt{{{- 2}{er}_{\max}^{2}} + {2\gamma\;{Mr}_{\max}} - L^{2}}} \end{matrix}}{r_{\max} \cdot r_{d} \cdot K}}},$ descending time ${t_{d} = {\frac{\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}} - \sqrt{{{- 2}{er}_{\max}^{2}} + {2\gamma\;{Mr}_{\max}} - L^{2}}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{\max}} - {\gamma\; M}}{\sqrt{K}}} - {{arc}\;\sin\frac{{2{er}_{d}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},$ descending latitude ${{DLa} = {{MLa} - {{\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {arc}}\;{\sin\left( {C \cdot R_{d}} \right\rbrack}}}},$ descending longitude ${{DLo} = {{MLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{d}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{DLa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{MLa}{2}} \right)}}} - {\Omega \cdot t_{d}}}},$ descending speed ${V_{d} = \sqrt{V_{\min}^{2} + {\frac{2\gamma\; M}{r_{d} \cdot r_{\max}} \cdot \left( {r_{\max} - r_{d}} \right)}}};$ launching angle ϑ_(o)≤0: descending radial coordinate r_(d)=R_(o):r_(i):(−1m), descending R-function ${R_{d} = {L \cdot \frac{\begin{matrix} {{\left( {{\gamma\;{Mr}_{o}} - L^{2}} \right)\sqrt{{{- 2}{er}_{d}^{2}} + {2\;\gamma\;{Mr}_{d}} - L^{2}}} -} \\ {\left( {{\gamma\;{Mr}_{d}} - L^{2}} \right)\sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}} \end{matrix}}{r_{o} \cdot r_{d} \cdot K}}},$ descending time ${t_{d} = {\frac{\sqrt{{{- 2}{er}_{d}^{2}} + {2\gamma\;{Mr}_{d}} - L^{2}} - \sqrt{{{- 2}{er}_{o}^{2}} + {2\gamma\;{Mr}_{o}} - L^{2}}}{2e} + {\frac{\gamma\; M}{2e\sqrt{2e}}\left( {{\arcsin\frac{{2{er}_{o}} - {\gamma\; M}}{\sqrt{K}}} - {{arc}\;\sin\frac{{2{er}_{d}} - {\gamma\; M}}{\sqrt{K}}}} \right)}}},$ descending latitude ${{DLa} = {{MLa} - {{\frac{\cos\;\beta}{{\cos\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot {arc}}\;{\sin\left( {C \cdot R_{d}} \right\rbrack}}}},$ descending longitude ${{DLo} = {{MLo} + {{\frac{\sin\; 2\;\beta}{{\sin\; 2\;\beta}} \cdot \frac{180{^\circ}}{\pi} \cdot \frac{\sqrt{1 - C^{2}}}{C \cdot \sqrt{1 - R_{d}^{2}}}}\ln\frac{{\tan\left( {{45{^\circ}} - \frac{DLa}{2}} \right)}}{{\tan\left( {{45{^\circ}} - \frac{MLa}{2}} \right)}}} - {\Omega \cdot t_{d}}}},$ descending speed ${V_{d} = \sqrt{V_{o}^{2} + {\frac{2\;\gamma\; M}{r_{d} \cdot r_{o}} \cdot \left( {r_{o} - r_{d}} \right)}}};$ outputting predicted projectile flight parameters without necessity of its monitoring by printing trajectory radial, longitudinal and latitudinal coordinates together with projectile velocities at corresponding times and plotting 3-D trajectory marking time and speed at desired locations. 